a solution ϕ0\phi_0 to the Euler-Lagrange equations of motion is said to exhibit spontaneously broken symmetry if it is not a fixed-point of that group action: if there is g∈Gg \in G such that g(ϕ0)≠ϕ0g(\phi_0) \neq \phi_0.

The “breaking” refers to the fact that the group no longer acts. It is called “spontaneous” because one imagines that by a physical process the theory “finds” one of its solutions. This comes from the class of examples where a statistical system is first considered at high temperature and then cooled down. At some point it will “spontaneously” freeze in one allowed configuration. A standard example is a ferromagnet?: at high temperature its magnetization? vanishes, while at very low termperature it spontaneously finds a direction of magnetization, thus “breaking” rotational symmetry.

One calls the subgroup Gϕ0⊂GG_{\phi_0} \subset G that fixes the given configuration ϕ0\phi_0 the unbroken symmetry group .

The fields in this effective QFT are then small excitations δϕ\delta \phi around the given ϕ0\phi_0. Since the original symmetry group still acts on the full fields ϕ0+δϕ\phi_0 + \delta \phi, the remaining symmetry group of the effective field theory is Gϕ0G_{\phi_0}, whose elements gg send

Since in the effective theory around ϕ0\phi_0 the vacuum state where all the δϕ\delta \phi have no excitations (or rather: are in their ground state) corresponds to ϕ0\phi_0 itself one says in this context that a quantum theory exhibits spontaneouly broken symmetry if its vacuum state is not invariant under the pertinent symmetries .

Formalization in cohesive homotopy-type theory

We indicate the formalization of the concept in the axiomatics of cohesion.

So this is something defined on phase space PP. If that also descends to the homotopy quotientP/GP/G (this is hard to draw here, but should be clear) then that makes the wavefunction also GG-equivariant. If not, then the wavefunction Ψ\Psi “breaks” the GG-symmetry.

Now if on top of this we have that the given Ψ\Psi is a “ground state”, then if it does not descend to the homotopy quotient we say “the GG-symmetry is spontaneously broken”.

To axiomatize what “ground state” means: introduce another ℝ\mathbb{R}-action on PP which is a Hamiltonian action, i.e. with respect to which the prequantum bundle is required to be equivariant. Then ask Ψ\Psi to (be polarized and) be a minimal eigenstate of the respective Hamiltonian. That makes it a “ground state”.

If both gg and hh are positive, then there is only one such critical point, given by Φ=0\Phi = 0. Therefore in this case the unique constant solution does not break the symmetry, in that g(Φ=0)=(Φ=0)g( \Phi = 0) = (\Phi = 0) for all g∈O(N)g \in O(N).

However, if hh is negative and gg positve, then the solutions are all those Φ\Phi with

|Φ|2=−hg>0.
\vert \Phi \vert^2 = - \frac{h}{g} \gt 0
\,.

The set of all these is closed under the action of G=O(N)G = O(N) – this group takes one of these solutions into another – but none of these solutions is fixed by this action.

One says in this case that any such solution ϕ:x↦Φ\phi : x \mapsto \Phi is a solution that spontaneously breaks the symmetry of the theory.

Kaluza-Klein reductions

Spontaneous symmetry breaking in gravity plays a central role for instance in the context of the Kaluza-Klein mechanism. For instance for dimX=5dim X = 5 the effective field theory of gravity around a solution of the form (X=X4×S1,g4⊗g1)(X = X_4 \times S^1, g_4 \otimes g_1) is 4-dimensional gravity coupled to electromagnetism (and a dilaton field): the components of the field of gravity along the circle transmute into the electromagnetic field. The ansatz breaks all the symmetries that would mix the remaining 4-dimensional gravitational excitations with these new electromagnetic excitations.

In this case, though, there are also supersymmetry analogs of the plain diffeomorphism action. Such a local supersymmetry remains unbroken in the given solution if it comes from a Killing spinor field.

points out that for symmetric systems with a symmetric ground state, already a tiny perturbation mixing the ground state with the first excited stated causes spontaneous symmetry breaking in the suitable limit, and suggests that this already resolves the measurement problem in quantum mechanics.